The Melvin Universe in Born-Infeld Theory and other Theories of Non-Linear Electrodynamics

15 Pages·0.2 MB·English

by  G. W. Gibbons

The Melvin Universe in Born-Infeld Theory
and other Theories of Non-Linear Electrodynamics
G W Gibbons∗ & C A R Herdeiro †
(D.A.M.T.P.)
1
0 Department of Applied Mathematics and Theoretical Physics,
0
Centre for Mathematical Sciences,
2
University of Cambridge,
n
a Wilberforce Road,
J
Cambridge CB3 0WA,
1
3 U.K.

1
January 2001
v
9
2
2
1 DAMTP-2001-9
0
1
Abstract
0
/
h We deriveaMelvinUniversetype solutiondescribinga magneticfieldpermeatingthe whole
-t Universeingravityminimallycoupledtoany non-linearelectromagnetictheory,includingBorn-
p
Infeld Theory. For a large set of non-linear electrodynamics theories, our solution is complete
e
h andnon-singular,aslongasthemagneticfieldissub-critical. Weexaminesomepropertiesofthe
: solution;inparticularthereisashiftofthe symmetryaxisandanon-standardperiodalongthe
v
i orbits of the U(1) symmetry to avoid a conical singularity. We show these are consistent with
X
the usualDiracquantizationconditionforthemagneticflux. We findexactsolutionsdescribing
r
a propagation of waves in the ‘generalized Melvin Universe’ along the principal null directions of
the electromagnetic field, where the BoillatandEinstein light-conetouch. By electric-magnetic
duality we show that similar Melvin electric and dyonic universes can be obtained.
1 Introduction
The‘Melvin magnetic universe’is a simplesolution to Einstein-Maxwell theory (or the bosonicsec-
tor of N=2 Supergravity) describing a bundle of magnetic flux lines in gravitational-magnetostatic
equilibrium [1]1. Subsequent work has generalized the solution in different directions, including
rotating and time dependent magnetic universes [3], black hole spacetimes which are asymptoti-
cally Melvin [4], gravitational waves travelling in the magnetic universe [5] and a Melvin universe
with a non-trivial axion and dilaton in heterotic string theory [6]. Particularly interesting is the
∗Email address: [email protected]
†Email address: [email protected]
1See also comments in [2] referring to work previousto Melvin’s on thesame solution.
1
Melvin Universe solution in Kaluza-Klein theory since it derives from a flat spacetime by perform-
ing U(1) Kaluza-Klein reduction with a twist in the identifications [20]. Recently [18] a version of
the Kaluza-Klein Melvin Universe was derived from M-theory as a solution to either type IIA or
type 0A string theory.
One of the most interesting applications of the Melvin Universe is the study of black hole pair
creation in a magnetic field. Although this spacetime configuration is stable against arbitrarily
large radial perturbations [7], it is unstable against the non-perturbative process of black hole pair
creation. This process was first discussed in [8], based on the ‘asymptotically’ Melvin ‘C-metric’
type solution found by Ernst [9]. The Euclideanized version of this metric is in fact the instanton
describing the creation in the Melvin Universe of a pair of magnetically charged extreme Reissner-
Nordstro¨m black holes with opposite charges [10]. In the Kaluza-Klein cases [21], a second decay
mode is by bubble nucleation. Moreover, and quite strikingly, the Ernst instanton is nothing else
than the Kerr solution in the higher dimension [21]. In M-theory the instability produces pairs of
D6-branes and anti-branes in the type IIA case, whereas for the type 0A the spacetime decays via
bubble nucleation.
In this note we generalise the Melvin spacetime in yet a different direction - as a solution to
gravity minimally coupled to any theory of non-linear electrodynamics (NLE) in four spacetime
dimensions. Our approach relies on a limiting procedure in Einstein-Maxwell theory which trans-
forms the Reissner-Nordstr¨om solution into the Melvin solution. Since the Reissner-Nordstr¨om
solution is known for an arbitrary theory of non-linear electrodynamics (see [11] which includes the
original references), it is straightforward to generalise the procedure to Einstein-NLE and hence
find the generalized Melvin solution.
Ourmotivation stemsbothfromthefactthatitmighthelpusunderstandphenomenalikeblack
holepaircreation ingravity coupledtoNLE,mostnotably Born-Infeldtheory, andinconnection to
ourrecentwork oncausality instringtheoryandNLE[12]. Thereinweshowed thattheopenstring
metric that has recently arisen in the work of Seiberg and Witten [13] is just the Boillat metric [14]
describing the propagation of fluctuation of the electromagnetic field in a non-trivial background
electromagnetic field in NLE. The Boillat metric is not, in general, conformal to the Einstein or
closed stringmetric. Hence, weexpectlightandgravitons totravel atdifferentspeeds,exceptalong
the principal null directions of the background electromagnetic field [12]. The generalized Melvin
solution we obtain in this paper will allow us to give examples of gravitational waves travelling in
a curved spacetime along the direction in which the Boillat and Einstein light-cone coincide.
This paper is organised as follows. In section 2 we describe the formalism of non-linear electro-
dynamics coupled to gravity and the spherically symmetric charged solution which will beessential
for our derivation. In section 3 we describe the limiting procedure and derive the solution. In
section 4 we discuss some properties of the solution and compare it to the Maxwell case. In par-
ticular, we discuss the existence of curvature singularities beyond the critical field in Born-Infeld
theory. A general shift of the symmetry axis is found. In order to avoid conical singularities a
non-standard period must be imposed along the orbits of the azimuthal Killing vector field. We
also discuss quantization of magnetic flux, showing that the standard condition is still obtained. In
section 5 we find exact solutions describing gravitational waves in the generalized Melvin Universe.
In section 6 we use electro-magnetic duality to derive electric and dyonic Melvin Universes. We
close with a discussion.
2
2 General Relativity coupled to non-linear electrodynamics
2.1 General Framework
If we consider gravity minimally coupled to non-linear electrodynamics, the action is
1 1
S = d4x√ gR+ d4x√ gL(x,y). (2.1)
16πG − 4π −
N
Z Z
Thequantities x 1F Fµν andy 1F ⋆Fµν arethetwoonlyindependentrelativisticinvariants
≡ 4 µν ≡ 4 µν
constructed from the Maxwell field in four spacetime dimensions, and L(x,y) an arbitrary function
of those variables. We will take Newton’s constant G = 1.
N
TheequationsofmotionderivedfromthisactionaremoreeasilywrittenintermsoftheLegendre
dual description of non-linear electrodynamics (see, for instance, [12]). This involves introducing
the tensor P defined by
µν
1
dL = PµνdF . (2.2)
µν
−2
P coincides with F for Maxwell’s theory. In general it reads
µν µν
P = (L F +L ⋆F ), (2.3)
µν x µν y µν
−
wheresubscriptson L denotedifferentiation. Thecomponents of P arejustDandH, theelectric
µν
induction and magnetic field respectively and hence (2.3) are faced as the constitutive relations for
thenon-linear electromagnetic theory. We thendenotebys andtthetwoindependentinvariants in
terms of the dual variables, i.e., s 1P Pµν and t 1P ⋆Pµν. The covariant Hamiltonian
≡ −4 µν ≡ −4 µν
Lˆ,
1
Lˆ = PµνF L, (2.4)
µν
−2 −
is a function of (s,t). In an orthonormal frame, Lˆ coincides with the usual Hamiltonian for purely
electric configurations and minus the usual Hamiltonian for purely magnetic ones. The equations
of motion for the coupled system are [17]
1
D Pµν = 0, R Rg = 8πT , (2.5)
µ µν µν µν
− 2
with the energy momentum tensor2
1
T = Lˆ P gαβP +g 2sLˆ +tLˆ Lˆ . (2.6)
µν s µα νβ µν s t
4π −
(cid:16) (cid:16) (cid:17)(cid:17)
2.2 Spherically symmetric solution
The spherically symmetric solution has been known since the 1930’s. One finds that Birkhoff’s
theorem holds. For a general theory, the electrically charged solution may be cast in the form (see
[11] which includes the original references)
−1
2m(r) 2m(r)
2 2 2 2 2 2 2
ds = 1 dt + 1 dr +r (dθ +sin θdφ ),
Einstein
− − r − r
(cid:18) (cid:19) (cid:18) (cid:19)
(2.7)
Qdt dr
P = ∧ .
r2
2We usea mostly plus signature. The Maxwell Lagrangian is L=−x=s.
3
The latter is the radial component of the electric induction D. The only non-trivial invariant is
then
2 4
s =Q /(2r ). (2.8)
The metric function that defines the solution obeys the differential equation
dm(r)
= r2Lˆ(s), (2.9)
dr
for s as in (2.8). These equations permit one to read off the behaviour of the solutions given the
function Lˆ(s). But it is actually easier to implement another program: choose m(r) such that
the geometry is non-singular and find the ‘theory’ that gives rise to such non-singular solution.
This program has been implemented in some recent papers [16]. Therein, possibilities for m(r) are
considered so as to approach the usual Reissner-N¨ordstrom solution at large r. Furthermore, the
geometry is chosen as to yield no curvaturesingularities; all scalar polynomials in the curvature are
bounded for small r. One then finds the theory for which such geometries are solution via (2.9).
Although one may construct interesting solutions in this fashion, resembling Bardeen tunnels, they
derive from obscure theories of non-linear electrodynamics.
3 The Melvin Universe
We now derive the generalized Melvin Universe solution to non-linear electrodynamics coupled to
gravity. Westartbydescribingalimitingprocedurethatgenerates theMelvinUniverseofEinstein-
Maxwell theory from the Reissner-N¨ordstrom solution. Then we apply the same procedure to the
non-linear electrodynamics case, and verify the solution by solving the Einstein equations. The
fields we obtain take the form:
ρ2+ 4 h(ρ) ρ2Λ(ρ)2
ds2 = H2 dφ2+ dρ2+Λ(ρ)2 dT2+dZ2 ,
Λ(ρ)2 ρ2+ 4 h(ρ) −
H2
(cid:0) (cid:1) (3.1)
Hρ
P = dφ dρ,
−Λ(ρ)2 ∧
where the functions h(ρ) and Λ(ρ) are given by
h(ρ) = 1 s−14 sLˆ(s′)s′−47ds′ , Λ(ρ) = 1+ H2ρ2. (3.2)
− 4 4
" Z0 #s=− H2
2Λ(ρ)4
For Maxwell’s theory of electromagnetism, Lˆ = s and so one may easily check that h(ρ) = 0. The
expression for h(ρ) might seem ill-defined, since it involves quartic roots that should be evaluated
at anegative quantity. Itshouldbeunderstood,however, thatfirstone shouldevaluate theintegral
and only then replace s by its value. In fact the expression is just a restatement of the infinite sum
(3.13). The constant H in the solution defines the magnetic field strength at the origin, which is,
in general, no longer at ρ= 0 as shown below.
4
3.1 Melvin Universe in Einstein-Maxwell theory
LetusdescribethelimitingprocedurethatgeneratestheMelvinuniversestartingwiththeReissner-
N¨ordstrom (RN) solution. Start by writing the RN solution usingstereographic coordinates for the
S2 sections:
4r2 2M Q2
ds2 = Vdt2+V−1dr2+ dζdζ¯, V 1 + ,
− (1+ζζ¯)2 ≡ − r r2
(3.3)
Q
F = dr dt,
−r2 ∧
Perform the diffeomorphism given by
ζ t
ζ˜= , r˜= rλ, t˜= , (3.4)
λ λ
to get the RN solution in (t˜,r˜,ζ˜,ζ¯˜) coordinates. Next we want to take the following limit:
3 2
λ 0, M,Q , Mλ = fixed m, Qλ = fixed q. (3.5)
→ → ∞ ≡ ≡
That the resulting spacetime configuration remains a solution requires proof. We will come back to
this point below. Notice that in the limit the S2 is ‘flattened’ into R2. This changes the symmetry
from spherical to axial. Denote ζ˜= x+iz and perform the final coordinate transformations and
relabellings:
2 T Z B2
t˜= iφ , x= i , z = , r˜= 1+ ρ2 Λ(ρ),
B2 2 2 4 ≡
(3.6)
2
q = iB, m = B /2.
−
Notice the change from timelike to spacelike character of t˜. This will change the electromagnetic
field from electric to magnetic. The resulting configuration is the Melvin Universe [1]:
ds2 = ρ2Λ(ρ)−2dφ2+Λ(ρ)2 dT2+dZ2+dρ2 ,
−
(3.7)
(cid:2) (cid:3)
F = BρΛ(ρ)−2dρ dφ.
∧
The constant B defining the solution is the magnetic induction strength at the origin, since B =
BΛ(ρ)−2∂z.
3.2 Generalized Melvin Universe
3.2.1 The Limiting Procedure
Now consider gravity coupled to general non-linear electrodynamics as described by (2.1). The
generalized RN solution is (2.7). It takes the form (3.3) with
2M I(r−1)
V = 1 + , (3.8)
− r r2
5
and F P the dual Maxwell tensor. Equation (2.9) translates into
→
dI(r−1) Q2
r−2 +r−1I(r−1)= 2r3Lˆ . (3.9)
d(r−1) 2r4
(cid:18) (cid:19)
for the function I(r−1). Next perform (3.4). In order to solve the differential equation we expand
I(λr˜−1) in a power series (around spatial infinity) and Lˆ(s) in its Taylor series around zero (we are
assuming Lˆ(s) to be analytic; we will be able to relax this assumption in the final result):
∞ ∞ sm ∂mLˆ(s)
I(λr˜−1)= Q2 f (λr˜−1)n, Lˆ(s)= . (3.10)
n m! ∂sm
" #
n=0 m=1 s=0
X X
RequiringthatLˆ approachthevalueforMaxwell’stheoryintheweakfieldlimityields[∂Lˆ/∂s]s=0 =
1 and for the usual RN solution to hold one must have f0 = 1. Substituting in (3.9) and matching
powers of λr−1 we find
∞ 1 Q2λ4 n ∂n+1Lˆ(s)
I(λr˜−1) = Q2 . (3.11)
(1+n)!(1+4n) 2r˜4 ∂sn+1
" #
n=0 (cid:18) (cid:19) s=0
X
Taking the limit (3.5), which for the function I yields
∞ 1 q2 n ∂n+1Lˆ(s)
lim λ4I(λr˜−1)= q2 . (3.12)
λ→0,Q→∞,λ2Q=q (1+n)!(1+4n) 2r˜4 " ∂sn+1 #
n=0 (cid:18) (cid:19) s=0
X
Finally perform (3.6) with relabelling B H to obtain the generalized Melvin magnetic universe
→
(3.1). The function h(ρ) is
∞ 1 H2 n ∂n+1Lˆ(s)
h(ρ) = . (3.13)
− (1+n)!(1+4n) −2Λ(ρ)4 ∂sn+1
" #
n=1 (cid:18) (cid:19) s=0
X
It can also be expressed in the more useful integral form (3.2).
3.2.2 Solving the Einstein equations
As we mentioned above, it is not clear that after performing the limit (3.5) one ends up with a
solutionoftheequationsofmotion. TheMaxwellequationin(2.5)isclearly obeyedby(3.1)sinceit
is completely identical to the usual Melvin solution and the determinant of the generalized Melvin
metricisunchangedfromtheusualMelvin solution. SowejustneedtoshowtheEinsteinequations
are solved. The Einstein tensor for the geometry (3.1) with Λ(ρ) given by (3.2) but unspecified
h(ρ) is diagonal and so is the energy-momentum tensor. The ρρ and φφ equations of motion are
equivalent, yielding the first order ordinary differential equation (O.D.E.)
dh(s)
4s2 s+sh(s) = Lˆ(s). (3.14)
ds − −
We are using the form of the only non-vanishing invariant
H2
s = , (3.15)
−2Λ(ρ)4
6
toexpressh(ρ)asafunctionofs. Itiseasytoverifythath(ρ)givenby(3.2)(withoutspecifyingthe
value of s in terms of ρ) is the solution. Notice that this equation does not define the lower limit in
the integral (3.2); this is set by the requirementthat one shouldrecover the usualMelvin spacetime
when Lˆ to correspond to Maxwell’s theory. We can therefore drop the analyticity assumption for
Lˆ(s). The tt and zz equations are also equivalent, yielding the second order O.D.E.
d2h(s) dh(s)
8s3 14s2 +s(1 h) = 2sLˆ Lˆ. (3.16)
− ds2 − ds − s−
Thisequation isnotindependentofthefirstorderO.D.E.(3.14). Usingthelatter tofindanexpres-
sion for Lˆ and replacing this in the second order O.D.E. one recovers thefirstorder equation. This
s
is similar to what happens for solution (2.7). Equation (2.9) comes from the tt or rr components
of the Einstein equations, whereas the θθ and φφ components give a second order equation which
is obtained by differentiating (2.9). Hence we have just shown that indeed the fields (3.1) are a
solution to the equations (2.5).
4 Properties of the solution and special cases
We now discuss some properties of the solution (3.1). We will enphasize two particular types of
theories of non-linear electrodynamics:
i) The family described by the Lagrangian:
a b
2 2
L = x+ x + y , (4.1)
− 2 2
which has a dual Lagrangian (up to terms of higher order in the constants a,b),
a b
Lˆ = s s2 t2. (4.2)
− 2 − 2
The Euler-Heisenberg theory (see for instance [19]) is the special case with a = 8α2/(45m4)
and b = 14α2/(45m4), where m is the electron mass and α the fine structure constant. The
function h(ρ) in (3.2) becomes
aH2
h(ρ) = . (4.3)
−20Λ(ρ)4
ii) The Born-Infeld Lagrangian:
1
L = 1 1+2β2x β4y2 , (4.4)
β2 − −
(cid:16) p (cid:17)
which has an exact dual Lagrangian of the same form
1
Lˆ = 1+2β2s β4t2 1 . (4.5)
β2 − −
(cid:16)p (cid:17)
In string theory β = 2πα′. The integral (3.2) does not have a closed form and the best one
can do in this case is to re-express h(ρ) in terms of the infinite sum (3.13):
∞ 1 H2β2 n n
h(ρ) = (2m 1) . (4.6)
− (1+n)!(1+4n) 2Λ(ρ)4 −
" #
n=1 (cid:18) (cid:19) m=1
X Y
Notice that both (4.3) for the Euler-Heisenberg case and (4.6) are negative for all values of ρ.
7
4.1 Singularities and asymptotics
The Einstein-Maxwell Melvin Universe (3.7) represents a parallel bundle of magnetic flux held
together by its own gravitational attraction. It is a totally non-singular configuration. It is easy to
check the absence of singularities in scalar polynomials of the curvature:
R = 0, R Rµν = 4B4Λ(ρ)−8, R Rµναβ = 4B4Λ(ρ)−8(3B4ρ4 24B2ρ2+80). (4.7)
µν µναβ
−
ThatR = 0followsfromtheconformalinvarianceofMaxwell’stheoryinfourspacetimedimensions.
The non-singular nature of the solution motivated Melvin to dub his solution a magnetic ‘geon’.
For the generalised Melvin solution, the existence of singularities will obviously depend on the
choice of Lagrangian. From the Einstein equations (2.5) one reads for the Ricci scalar
dLˆ(s)
R = 8 Lˆ(s) s , (4.8)
− ds
" #s=− H2
2Λ(ρ)4
sothatiftheenergydensityoritsgradientblowsupsodoesthecurvature. FortheEuler-Heisenberg
family this curvature invariant is bounded everywhere, but for the Born-Infeld case it diverges at
radial distance
4 2 2
Λ(ρ) = β H . (4.9)
Therefore, if the constant H in the solution is such that H2 > β−2, then the solution is singular.
The value H2 = β−2 is the maximal magnetic field for a purely magnetic configuration in Born-
Infeld theory. Notice that Born-Infeld theory has a maximal value for the electric field E~ and
for the magnetic field H~ (but not for the inductions). Thus, the free parameter describing the
solution is constrained if we require absence of curvature singularities. In general, we expect that if
the non-linear theory of electrodynamics under consideration has a critical field the curvature will
diverge at that critical field.
Independently of the existence of curvature singularities, the metric (3.1) will be singular if
there exists a radial distance ρ0 such that
4
2
ρ0+ H2h(ρ0)= 0. (4.10)
This is certainly the case for Euler-Heisenberg and Born-Infeld, since h(ρ) is negative everywhere
for these two theories. Consider the ρ-φ sections of the metric (3.1)
ρ2
2 2 2
ds = dρ +g dφ . (4.11)
φφ
g
φφ
In the neighbourhood of ρ = ρ0,
ρ2
2 0 2 2
ds dρ +g˙φφ(ρ ρ0)dφ , (4.12)
≃ g˙φφ(ρ ρ0) −
−
where g˙φφ = (dgφφ/dρ) at ρ = ρ0, which we assume to be non-zero. Now introduce the new
coordinate R by
R2
(ρ−ρ0) = 4g˙φφ ρ2 , (4.13)
0
8
to get the metric
2g˙
2 2 2 φφ
ds dR +R d φ . (4.14)
≃ ρ0
(cid:18) (cid:19)
Hence, if we identify φ with period
πρ0
δφ = , (4.15)
g˙
φφ
the singularity at ρ = ρ0 is just the usual singularity at the origin of flat space in polar coordinates.
Any other period for φ leads to a conical singularity. In any case, the point ρ = ρ0 should be faced
as the location of the axis of symmetry, where ρ0 is the first solution of (4.10) when coming from
infinity. The region 0 < ρ < ρ0 will be discussed below.
Asymptotically, ourgeneralized Melvin Universewillapproach theusualMelvin solution, which
is clearly a consequence of the requirement that the theory of non-linear electrodynamics under
consideration should approach Maxwell’s theory for weak enough fields. The components of the
Riemann tensor in an orthonormal frame vanish as 1/ρ6. In this sense, asymptotic Melvin space-
timesareasymptotically flat(away fromtheaxisofsymmetry). ThecomponentsoftheWeyltensor
in an orthonormal frame, Ca , also vanish with the same power of ρ. Thefact that this is truefor
bcd
all non-vanishing components indicates that there might exists a conformal compactification of the
spacetime, which however is not know. If it exists the conformal spacetime should have a light-like
conformal boundary.
We also note that the isometry group of thesolution (3.1) is the same as the one of (3.7). There
is a U(1) ISO(1,1) isometry corresponding to a circle action along the φ direction, boosts along
×
the t-z direction and translations along the t and z directions.
4.2 Test motions and supersymmetry
In the Newtonian limit, the generalised Melvin Universe has a gravitational potential equal to the
usual Melvin solution, which is of the form
H2 H4
2 4
V = ρ + ρ . (4.16)
4 32
This potential well will tend to confine the particles in a cylinder of finite radius. In this respect
the behaviour is similar to that of Anti-de-Sitter spacetime, despite the fact that AdS has constant
curvature in contrast to the asymptotically vanishing curvature of Melvin discussed in the previous
subsection. For the full relativistic analysis consider the geodesic equations, where λ is an affine
parameter along the trajectory:
dρ 2 ρ2+4H−2h(ρ) 1 l2
2 2 2
= (E p ) m ,
dλ ρ2Λ(ρ)2 Λ(ρ)2 − − − ρ2
(cid:18) (cid:19) (cid:20) (cid:21)
(4.17)
dφ lΛ(ρ)2 dt E dz p
= , = , = .
dλ ρ2+4H−2h(ρ) dλ Λ(ρ)2 dλ Λ(ρ)2
The conserved quantities E,p,l are associated with the Killing vector fields ∂/∂t,∂/∂z and ∂/∂φ
respectively. The‘mass’misasusualassociatedwiththemetricKillingtensor. Thet,z coordinates
arethepropertimeandproperradialdistanceforanobserverattheorigin. Asseenbythisobserver
9
clocks tick faster as the radial distance increases. This is exactly the opposite of the redshift
witnessed by the observer at infinity for events close to the black hole horizon in a Schwarzchild
spacetime.
From the radial equation we can see that at large distances the repulsive terms dominate over
the attractive (energy) term. Hence, no particles may enter the large enough ρ region, whatever
values the particle’s quantum numbers take. The solely exception is a photon with no angular
momentum not moving entirely in the z direction, which is not confined to a cylinder of some
radius. In this way we see that the Newtonian intuition is mostly confirmed. This resembles the
Melvin case [15] due to the similar asymptotics.
The behaviour for small ρ is quite distinct from the usual Melvin solution. From the radial
equation we can see that no particle can cross the ρ = ρ0 cylinder. But geodesics may exist on
both sides of this cylinder. As an example consider the Euler-Heisenberg case. Equation (4.10)
only has one solution. Its left hand side is negative in 0 < ρ < ρ0 and positive for ρ > ρ0. Bound
states with E2 < m2 can therefore exist within the ρ = ρ0 cylinder. In a sense this is analogous to
the r < 0 region in the Kerr metric. Although ρ = ρ0 should be seen as the location of the axis
of symmetry (like the r = 0 in Kerr), there is a region beyond it for which the metric is different
and therefore taking it seriously is not double counting (like the negative ρ region for flat space in
polar coordinates). Hence the spacetime can have two geodesically complete disjoint regions.
To close this section we remark that on a curious effect for the Einstein-Maxwell Melvin Uni-
verse: at any radial distance one may find a ‘no-force’ configuration, i.e. a static test particle. The
charged geodesics are
dρ 2 1 m2 1 2q 2
2 2
= (E p ) l ,
dλ Λ(ρ)4 − − Λ(ρ)2 − ρ2 − BΛ(ρ)
(cid:18) (cid:19) (cid:18) (cid:19)
(4.18)
dφ Λ(ρ)2 2q
= l ,
dλ ρ2 − BΛ(ρ)
(cid:18) (cid:19)
where q is the particle’s charge. The t and z equations are the same as in (4.17). Notice that the
charged geodesics for the generalised Melvin Universe are more subtle due to the coupling to the
Maxwell potential. Choosing l,q as to cancel the right hand side of the φ equation of motion (4.18)
for a given ρ, putting p = 0 to cancel z motion (4.17) and choosing E2 = Λ(ρ)2m2, the particle
becomes static. Obviously this cannot happen for photons (m = 0).
The possibility of a no-force configuration, together with the fact that the Reissner-N¨ordstrom
family includesasupersymmetricsolution raisesaquestionaboutthesupersymmetryoftheMelvin
solution withinN = 2, D = 4supergravity. TheKillingspinorequation is,inthenotation offorms,
1 1
d+ ω Γab F ΓabΓ ǫ = 0. (4.19)
ab ab
4 − 4
(cid:18) (cid:19)
We use for the flat metric η = diag( 1,1,ρ2,1), i.e., flat space in cylindrical coordinates. Then,
ab
−
one easily checks that the Killing spinor equation implies the conditions:
2
ǫ = Λ12ǫ0, 1 Γφ ǫ0 = 0, (4.20)
− B
(cid:18) (cid:19)
10